Natural language

The language of (∞,1)(\infty,1)-categories happens to naturally capture, unify and also simplify a plethora of constructions and considerations in homotopy theory, homological algebra and cohomology theory. All these subjects are thereby seen to be different realizations of a single underlying principle.

To appreciate this point, it may be useful to first consider the analogous statement in the case of 1-categories:

preparation: set theory is the theory of the category of sets

While mathematics is based to a large extent on the notion of a set, the search for the right formulation of set theory has been a long one. William Lawvere famously argued, in particular as discussed in his book

David : Just as not all reasoning in the topos Set carries over to other toposes, what can be said of the reasoning in Top that does/does not work in other (infinity, 1)-toposes?

Urs: good question. I added one little remark in response to this below. But likely much more could be said here (both known to date and not yet known to date, I suppose). In particular, this only addresses the geometric aspects of (∞,1)(\infty,1)-topoi. I don’t think anyone has as yet considered (∞,1)(\infty,1)-topoi as context in which to interpret logic. And after all, one must not forget that, HTT is only (hah!) about Grothendieck(∞,1)(\infty,1)-topoi ((∞,1)(\infty,1)-presheaf topoi).

side remark: relation to more general ∞\infty-categories

the problem of defining higher categories is the problem of defining the right coherence laws – associators, pentagonators and so forth. But even if the higher category in question is not groupoidal, i.e. is directed, the coherence cells will be equivalences, i.e. weakly invertible cells. So the problem of controlling coherence laws is a problem of (∞,0)(\infty,0)-categorical nature.

For this reason it is useful to first go to infinit cell degree with just the invertible cells, and only after that start increasing the degree of the non-invertible cells.

Concretely, we have the following useful constructions for (∞,1)(\infty,1)-categories:

there is a simple definition of the (infinity,1)-category of (infinity,1)-categories. This should be the (∞,1)(\infty,1)-subcategory on all invertible 2-cells (transformations of (∞,1)(\infty,1)-functors) of the (infinity,2)-category of (infinity,1)-categories?, which has a more involved definition, but the point is that the (∞,1)(\infty,1)-category of all (∞,1)(\infty,1)-categories is already quite sufficient for many constructions.

there is a simple iterative definition of (infinity,n)-category by an iterative weak enrichment in (∞,n−1)(\infty,n-1)-categories, again precisely due to the presence of invertible cells in all degrees, which allows to say what a homotopy limit of (∞,n−1)(\infty,n-1)-categories is, as enters for instance in the definition of complete Segal space.

new possibilities

Natural language is never just a value in its own right, but always the potential to achieve new things which are literally unthinkable without this language.

The identification of a coherent framework of (∞,1)(\infty,1)-categories is rapidly leading to a wealth of new developments in areas where infinity-category theory has long been expected to be crucial, but never quite lived up to the status of a useful well-developed tool that would allow to go beyond its own introspection.

Discussion

A previous version of this entry triggered the following discussion.

This gives impression that (infty,1) categories are the first higher categorical framework to naturally capture TQFT-s, and derived techniques in rep theory. My impression is that so far much greater number of concrete cases has been worked out using A-infty categories, related to mirror symmetry and related models; and the picture in Lagrangean geometry and homological mirror symmetry emerged in works of Fukaya and Kontsevich in early 1990s. Can (infty, 1) approach at least reproduce after the fact the results which can be studied using Ainfty setup ? in particular is there a good treatment of Lagrangean intersection theory using derived geometry of topos and (infty,1) school ? – Zoran

Urs: good point. I’ll have to think about that. Just two quick remarks:

David Ben-Zvi keeps emphasizing that the dg-categories used in much of this QFT business are naturally thought of as (∞,1)(\infty,1)-categories. For instance in his latest article he makes some connection to the Kapustin-Witten description.

Over on the blog John and Mike are talking about whether and in which sense A∞A_\infty-categories are models for (∞,1)(\infty,1)-categories, too.

David Ben-Zvi (by email, forwards here by Urs): the A−∞A-\infty categories appearing in the literature mentioned above are all A−∞A-\infty categories in the usual linear sense (unlike the nonlinear ones John talks about on the blog). Anyway these are all quasiequivalent to dg categories - ie. the ∞\infty-cats of dg cats and A∞A_\infty-cats are all the same. In particular they are all examples of (∞,1)(\infty,1)-cats – and are almost stable (ie. they might be too small, like dg cats with one object, but if you add cones to them – aka passing to their category of modules or derived category – you get a stable ∞\infty-cat. So it’s really the same subject, or rather rationally these are all equivalent languages.)